Extended Theories of Gravity
Extended Theories of Gravity can be considered a new paradigm to cure shortcomings of General Relativity at infrared and ultraviolet scales. They are an approach that, by preserving the undoubtedly positive results of Einstein’s Theory, is aimed to address conceptual and experimental problems recently emerged in Astrophysics, Cosmology and High Energy Physics. In particular, the goal is to encompass, in a self-consistent scheme, problems like Inflation, Dark Energy, Dark Matter, Large Scale Structure and, first of all, to give at least an effective description of Quantum Gravity. We review the basic principles that any gravitational theory has to follow. The geometrical interpretation is discussed in a broad perspective in order to highlight the basic assumptions of General Relativity and its possible extensions in the general framework of gauge theories. Principles of such modifications are presented, focusing on specific classes of theories like -gravity and scalar-tensor gravity in the metric and Palatini approaches. The special role of torsion is also discussed. The conceptual features of these theories are fully explored and attention is payed to the issues of dynamical and conformal equivalence between them considering also the initial value problem. A number of viability criteria are presented considering the post-Newtonian and the post-Minkowskian limits. In particular, we discuss the problems of neutrino oscillations and gravitational waves in Extended Gravity. Finally, future perspectives of Extended Gravity are considered with possibility to go beyond a trial and error approach.
- I Introduction
- II A summary of gauge symmetries
III Gravity from gauge invariants
- III.1 What can ”generate” gravity?
- III.2 Invariance Principles and the Noether Theorem
- III.3 The Global Poincaré invariance
- III.4 The Local Poincaré invariance
- III.5 Spinors, vectors, bivectors and tetrads
- III.6 Curvature, torsion and metric
- III.7 The field equations of gravity
- III.8 Concluding remarks
- IV Deformations and conformal transformations
- V The physical meaning of General Relativity
- VI Introduction
- VII Quantum field theory in curved space-time
- VIII Variational principles and field equations in metric formalism
- IX The Palatini formalism
- X Conformal transformations and Extended Theories of Gravity
- XI Extended Theories with torsion
XII The -bundles framework
- XII.1 The field equations in the -bundle formalism
- XII.2 -gravity within the -bundle framework
- XII.3 Symmetries and conserved quantities
- XII.4 -gravity in presence of matter
- XII.5 Equivalence with scalar-tensor theories
- XII.6 Discussion
XIII The Hamiltonian formulation
- XIII.1 The Hamiltonian constraint of General Relativity
- XIII.2 The Hamiltonian constraint for -gravity
- XIII.3 The cosmological constant as an eigenvalue
- XIII.4 The transverse traceless (TT) spin 2 operator for the Schwarzschild metric and the WKB approximation
- XIII.5 One-loop energy Regularization and Renormalization
- XIII.6 Concluding remarks
XIV The initial value problem
- XIV.1 The Cauchy problem of scalar-tensor gravity
- XIV.2 The initial value problem of -gravity in the ADM formulation
- XIV.3 The Gaussian normal coordinates approach in General Relativity
- XIV.4 The Cauchy problem of -gravity in Gaussian normal coordinates
XV Spherical and axial symmetry
- XV.1 Spherically symmetric solutions in -gravity
- XV.2 The Noether approach and spherical symmetry
- XV.3 From spherical to axial symmetry in -gravity
- XVI The Post- Newtonian limit
XVII The Post- Minkowskian limit
- XVII.1 The weak field limit in Minkowski space-time
- XVII.2 The -gravity energy-momentum pseudo-tensor and gravitational radiation
- XVII.3 Ghosts, massless and massive gravitational modes
- XVII.4 Concluding remarks
- XVIII Conclusions and Perspectives
Part I : Generalities and open problems
Several issues and shortcomings emerged in the last thirty years leading to the conclusion that Einstein’s General Relativity is not the final theory of gravitational interaction. The goal of this Report is to take into account some of these problems to show that more general approaches to gravity have to be pursued.
The Review article is organised in three parts. Each part has an introduction, a main body and is self-contained.
Part I is devoted to a general discussion of gravitational interaction under the standard of gauge theories. Our aim is to show the path that led to General Relativity, its self-consistency and successes in addressing open problem at Einstein’s time. A summary of shortcomings emerged later is given. In particular we discuss problems at infra-red and ultra-violet scales. After we review the Gauge Theory showing that General Relativity comes out from local gauge transformations, local Poincaré invariance and space-time symmetries. The role of space-time deformations and conformal transformations is discussed in view of Extended Theories of Gravity. We close this part by pointing out the physical meaning of General Relativity, in particular discussing the Equivalence Principle, the geodesic and metric structures, the post-Newtonian and post-Minkowskian limits.
Part II is the main part of this Report. After a general discussion of Extended Theories of Gravity, we provide a summary of their emergence in Quantum Field Theory formulated in curved space-time. We develop the variational principles and the field equations for some classes of Extended Theories in metric and Palatini formalism. Conformal transformations and their physical interpretation are widely discussed. The role of torsion in these theories is analysed. Besides, we develop their jet-bundle representation in order to put in evidence the role of symmetries and conserved quantities. After we present the Hamiltonian formulation showing how it can be connected to the problems of cosmological constant and renormalization at one-loop level. Finally the Initial Value Problem, in various formulations, is addressed showing that the well-formulation and well-position are still not available for any Extended Theory of Gravity.
Part III is devoted to the applications. We consider exact solutions in spherical and axial symmetry. After we develop post-Newtonian and post-Minkowskian limits. The main achievement is the fact that further features as corrections to Newton potential, new polarizations and new gravitational modes emerge with respect to General Relativity. In our opinion, these characteristics will be the test-bed capable of confirming or ruling out Extended Theories of Gravity.
We conclude the Report with some Appendices containing notations, zeta-function regularization and technicalities on Noether symmetries.
i.1 A short history of Theories of Gravity
It is remarkable that gravity is probably the fundamental interaction which still remains the most enigmatic, even though it is related to phenomena experienced in everyday life and is the one most easily conceived without any sophisticated knowledge. As a matter of fact, the gravitational interaction was the first one to have been put under the microscope of experimental investigation, obviously due to the simplicity of constructing a suitable experimental apparatus.
Galileo Galilei was the first to introduce pendula and inclined planes to the study of terrestrial gravity at the end of the 16th century galileo (); galileo1 (); galileo2 (); galileo3 (); galileo4 (). Gravity played an important role in the development of Galileo’s ideas about the necessity of experiment in the study of Science, which had a great impact on modern scientific thinking. However, it was not until 1665, when Isaac Newton introduced the now renowned ”inverse-square gravitational force law”, that terrestrial gravity was actually related to celestial gravity in a single theory newtonG (); newton1 (). Newton’s theory made correct predictions for a variety of phenomena at different scales, including both terrestrial experiments and planetary motion.
Obviously, Newton’s contribution to gravity, quite apart from his enormous contribution to physics overall, is not restricted to the expression of the inverse square law. Much attention should be paid to the conceptual basis of his gravitational theory, which incorporates two key ideas:
The idea of absolute space, i.e. the view of space as a fixed, unaffected structure; a rigid arena where physical phenomena take place.
The idea of what was later called the Weak Equivalence Principle which, expressed in the language of Newtonian theory, states that the inertial and the gravitational mass coincide.
Asking whether Newton’s theory, or any other physical theory, is right or wrong, would be an ill-posed question to begin with, since any consistent theory is apparently ”right”. A more appropriate way to pose the question would be to ask how suitable this theory is to describe the physical world or, even better, how large a portion of the physical world is sufficiently described by such a theory. Also, one could ask how unique the specific theory is for the description of the relevant phenomena. It was obvious, in the first 20 years after the introduction of Newtonian gravity, that it did manage to explain all of the aspects of gravity known at that time. However, all of the questions above were posed sooner or later.
In 1855, Urbain Le Verrier observed a 35 arc-second excess precession of Mercury’s orbit and later on, in 1882, Simon Newcomb measured this precession more accurately to be 43 arc-seconds lever (); lever1 (); newc (); mach (); mach1 (). This experimental fact was not predicted by Newton’s theory. It should be noted that Le Verrier initially tried to explain the precession within the context of Newtonian gravity, attributing it to the existence of another, yet unobserved, planet whose orbit lies within that of Mercury. He was apparently influenced by the fact that examining the distortion of the planetary orbit of Uranus in 1846 had led him, and, independently, John Couch Adams, to the discovery of Neptune and the accurate prediction of its position and momenta. However, this innermost planet was never found.
On the other hand, in 1893, Ernst Mach stated what was later called by Albert Einstein ”Mach’s principle”. This is the first constructive attack to Newton’s idea of absolute space after the 18th century debate between Gottfried Wilhelm von Leibniz and Samuel Clarke (Clarke was acting as Newton’s spokesman) on the same subject, known as the Leibniz–Clarke Correspondence alexander (). Mach’s idea can be considered as rather vague in its initial formulation and it was essentially brought into the mainstream of physics later on by Einstein along the following lines:
“…inertia originates in a kind of interaction between bodies…”.
This is obviously in contradiction with Newton’s ideas, according to which inertia was always relative to the absolute frame of space. There exists also a later, probably clearer interpretation of Mach’s Principle, which, however, also differs in substance. This was given by Dicke dicke ():
”The gravitational constant should be a function of the mass distribution in the Universe”.
This is different from Newton’s idea of the gravitational constant as being universal and unchanging. Now Newton’s basic axioms have to be reconsidered.
But it was not until 1905, when Albert Einstein completed Special Relativity, that Newtonian gravity would have to face a serious challenge. Einstein’s new theory, which managed to explain a series of phenomena related to non-gravitational physics, appeared to be incompatible with Newtonian gravity. Relative motion and all the linked concepts had gone well beyond Galileo and Newton ideas and it seemed that Special Relativity should somehow be generalised to include non-inertial frames. In 1907, Einstein introduced the equivalence between gravitation and inertia and successfully used it to predict the gravitational redshift. Finally, in 1915, he completed the theory of General Relativity (GR), a generalisation of Special Relativity which included gravity and any accelerated frame. Remarkably, the theory matched perfectly the experimental result for the precession of Mercury’s orbit, as well as other experimental findings like the Lense-Thirring LT (); LT1 (); LT2 (); LT3 () gravitomagnetic precession (1918) and the gravitational deflection of light by the Sun, as measured in 1919 during a Solar eclipse by Arthur Eddington eddington ().
GR overthrew Newtonian gravity and continues to be up to now an extremely successful and well-accepted theory for gravitational phenomena. As mentioned before, and as often happens with physical theories, Newtonian gravity did not lose its appeal to scientists. It was realised, of course, that it is of limited validity compared to GR, but it is still sufficient for most applications related to gravity. What is more, in weak field limit of gravitational field strength and velocities, GR inevitably reduces to Newtonian gravity. Newton’s equations for gravity might have been generalised and some of the axioms of his theory may have been abandoned, like the notion of an absolute frame, but some of the cornerstones of his theory still exist in the foundations of GR, the most prominent example being the Equivalence Principle, in a more suitable formulation of course.
This brief chronological review, besides its historical interest, is outlined here also for a practical reason. GR is bound to face the same questions as were faced by Newtonian gravity and many people would agree that it is actually facing them now. In the forthcoming sections, experimental facts and theoretical problems will be presented which justify that this is indeed the case. Remarkably, there exists a striking similarity to the problems which Newtonian gravity faced, i.e. difficulty in explaining several observations, incompatibility with other well established theories and lack of uniqueness.
i.2 What we mean for a ”Good Theory of Gravity”
From a phenomenological point of view, there are some minimal requirements that any relativistic theory of gravity has to match. First of all, it has to explain the astrophysical observations (e.g. the orbits of planets, self-gravitating structures).
This means that it has to reproduce the Newtonian dynamics in the weak-energy limit. Besides, it has to pass the classical Solar System tests which are all experimentally well founded Will93 (). As second step, it should reproduce Galactic dynamics considering the observed baryonic constituents (e.g. luminous components as stars, sub-luminous components as planets, dust and gas), radiation and Newtonian potential which is, by assumption, extrapolated to Galactic scales.
Thirdly, it should address the problem of large scale structure (e.g. clustering of galaxies) and finally cosmological dynamics, which means to reproduce, in a self-consistent way, the cosmological parameters as the expansion rate, the Hubble constant, the density parameter and so on. Observations and experiments, essentially, probe the standard baryonic matter, the radiation and an attractive overall interaction, acting at all scales and depending on distance: the gravity.
The simplest theory which try to satisfies the above requirements is the GR einstein (). It is firstly based, on the assumption that space and time have to be entangled into a single space-time structure, which, in the limit of no gravitational forces, has to reproduce the Minkowski space-time structure. Einstein profitted also of ideas earlier put forward by Riemann, who stated that the Universe should be a curved manifold and that its curvature should be established on the basis of astronomical observations riemann ().
In other words, the distribution of matter has to influence point by point the local curvature of the space-time structure. The theory, eventually formulated by Einstein in 1915, was strongly based on three assumptions that the Physics of Gravitation has to satisfy.
The ”Principle of Relativity”, that requires all frames to be good frames for Physics, so that no preferred inertial frame should be chosen a priori (if any exist).
The ”Principle of Equivalence”, that amounts to require inertial effects to be locally indistinguishable from gravitational effects (in a sense, the equivalence between the inertial and the gravitational mass).
The ”Principle of General Covariance”, that requires field equations to be ”generally covariant” (today, we would better say to be invariant under the action of the group of all space-time diffeomorphisms) schroedinger ().
And - on the top of these three principles - the requirement that causality has to be preserved (the ”Principle of Causality”, i.e. that each point of space-time should admit a universally valid notion of past, present and future).
Let us also recall that the older Newtonian theory of space-time and gravitation, that Einstein wanted to reproduce at least in the limit of weak gravitational forces (what is called today the ”post-Newtonian approximation”), required space and time to be absolute entities, particles moving in a preferred inertial frame following curved trajectories, the curvature of which (e.g. the acceleration) had to be determined as a function of the sources (i.e. the ”forces”).
On these bases, Einstein was led to postulate that the gravitational forces have to be expressed by the curvature of a metric tensor field on a four-dimensional space-time manifold, having the same signature of Minkowski metric, e.g., the so-called ”Lorentzian signature”, herewith assumed to be . He also postulated that space-time is curved in itself and that its curvature is locally determined by the distribution of the sources, e.g., being space-time a continuum, by the four-dimensional generalization of what in Continuum Mechanics is called the ”matter stress-energy tensor”, e.g. a rank-two (symmetric) tensor .
Hilbert and Einstein schroedinger () proved that the field equations for a metric tensor , related to a given distribution of matter-energy, can be achieved by starting from the Ricci curvature scalar which is an invariant. We will give details below.
The choice of Hilbert and Einstein was completely arbitrary (as it became clear a few years later), but it was certainly the simplest one both from the mathematical and the physical point of view. As it was later clarified by Levi–Civita in 1919, curvature is not a ”purely metric notion” but, rather, a notion related to the ”linear connection” to which ”parallel transport” and ”covariant derivation” refer levicivita ().
In a sense, this is the precursor idea of what in the sequel would be called a ”gauge theoretical framework” gauge (), after the pioneering work by Cartan in 1925 cartan (). But at the time of Einstein, only metric concepts were at hands and his solution was the only viable.
It was later clarified that the three principles of relativity, equivalence and covariance, together with causality, just require that the space-time structure has to be determined by either one or both of two fields, a Lorentzian metric and a linear connection , assumed at the beginning to be torsionless for the sake of simplicity.
The metric fixes the causal structure of space-time (the light cones) as well as its metric relations (clocks and rods); the connection fixes the free-fall, i.e. the locally inertial observers. They have, of course, to satisfy a number of compatibility relations which amount to require that photons follow null geodesics of , so that and can be independent, a priori, but constrained, a posteriori, by some physical restrictions. These, however, do not impose that has necessarily to be the Levi-Civita connection of palatiniorigin ().
This justifies - at least on a purely theoretical basis - the fact that one can envisage the so-called ”alternative theories of gravitation”, that we prefer to call ”Extended Theories of Gravitation” (ETGs) since their starting points are exactly those considered by Einstein and Hilbert: theories in which gravitation is described by either a metric (the so-called ”purely metric theories”), or by a linear connection (the so-called ”purely affine theories”) or by both fields (the so-called ”metric-affine theories”, also known as ”first order formalism theories”). In these theories, the Lagrangian is a scalar density of the curvature invariants constructed out of both and .
The choice of Hilbert-Einstein Lagrangian is by no means unique and it turns out that the Hilbert-Einstein Lagrangian is in fact the only choice that produces an invariant that is linear in second derivatives of the metric (or first derivatives of the connection). A Lagrangian that, unfortunately, is rather singular from the Hamiltonian point of view, in much than same way as Lagrangians, linear in canonical momenta, are rather singular in Classical Mechanics (see e.g. arnold ()).
A number of attempts to generalize GR (and unify it to Electromagnetism) along these lines were followed by Einstein himself and many others (Eddington, Weyl, Schrödinger, just to quote the main contributors; see, e.g., unification ()) but they were eventually given up in the fifties of XX Century, mainly because of a number of difficulties related to the definitely more complicated structure of a non-linear theory (where by ”non-linear” we mean here a theory that is based on non-linear invariants of the curvature tensor), and also because of the new understanding of physics that is currently based on four fundamental forces and requires the more general ”gauge framework” to be adopted (see unification2 ()).
Still a number of sporadic investigations about ”alternative theories” continued even after 1960 (see Will93 () and Refs. quoted therein for a short history). The search for a coherent quantum theory of gravitation or the belief that gravity has to be considered as a sort of low-energy limit of string theories green (), something that we are not willing to enter here in detail, has more or less recently revitalized the idea that there is no reason to follow the simple prescription of Einstein and Hilbert and to assume that gravity should be classically governed by a Lagrangian linear in the curvature.
Further curvature invariants or non-linear functions of them should be also considered, especially in view of the fact that they have to be included in both the semi-classical expansion of a quantum Lagrangian or in the low-energy limit of a string Lagrangian.
Moreover, it is clear from the recent astrophysical observations and from the current cosmological hypotheses that Einstein equations are no longer a good test for gravitation at Solar System, Galactic, extra-galactic and cosmic scale, unless one does not admit that the matter side of field equations contains some kind of exotic matter-energy which is the ”dark matter” and ”dark energy” side of the Universe.
The idea which we propose here is much simpler. Instead of changing the matter side of Einstein field equations in order to fit the ”missing matter-energy” content of the currently observed Universe (up to the of the total amount!), by adding any sort of inexplicable and strangely behaving matter and energy, we claim that it is simpler and more convenient to change the gravitational side of the equations, admitting corrections coming from non-linearities in the effective Lagrangian. However, this is nothing else but a matter of taste and, since it is possible, such an approach should be explored. Of course, provided that the Lagrangian can be conveniently tuned up (i.e., chosen in a huge family of allowed Lagrangians) on the basis of its best fit with all possible observational tests, at all scales (Solar, Galactic, extragalactic and cosmic).
Something that, in spite of some commonly accepted but disguised opinion, can and should be done before rejecting a priori a non-linear theory of gravitation (based on a non-singular Lagrangian) and insisting that the Universe has to be necessarily described by a rather singular gravitational Lagrangian (one that does not allow a coherent perturbation theory from a good Hamiltonian point of view) accompanied by matter that does not follow the behaviour that standard baryonic matter, probed in our laboratories, usually satisfies.
i.3 General Relativity and its shortcomings
Considering the above discussion it is worth noticing that in the last thirty years several shortcomings came out in the Einstein theory and people began to investigate whether GR is the only fundamental theory capable of explaining the gravitational interaction. Such issues come, essentially, from cosmology and quantum field theory. The shortcomings are related both to many theoretical aspects and to observational results. In this section we will try to summarize these problems. An important issue has to be underlined: even if there are many problems, the reaction of scientific community is not uniform. In a very simple scheme we can summarize the guide lines.
Many people will agree that modern physics is based on two main pillars: GR and Quantum Field Theory. Each of these two theories has been very successful in its own arena of physical phenomena: GR in describing gravitating systems and non-inertial frames from a classical point of view on large enough scales, and Quantum Field Theory at high energy or small scale regimes where a classical description breaks down. However, Quantum Field Theory assumes that space-time is flat and even its extensions, such as Quantum Field Theory in curved space time, consider space-time as a rigid arena inhabited by quantum fields. GR, on the other hand, does not take into account the quantum nature of matter. Therefore, it comes naturally to ask what happens if a strong gravitational field is present at quantum scales. How do quantum fields behave in the presence of gravity? To what extent are these amazing theories compatible?
Let us try to pose the problem more rigorously. Firstly, what needs to be clarified is that there is no final proof that gravity should have some quantum representation at high energies or small scales, or even that it will retain its nature as an interaction. The gravitational interaction is so weak compared with other interactions that the characteristic scale under which one would expect to experience non-classical effects relevant to gravity, the Planck scale, is cm. Such a scale is not of course accessible by any current experiment and it is doubtful whether it will ever be accessible to future experiments either 111This fact does not imply, of course, that imprints of Quantum Gravity phenomenology cannot be found in lower energy experiments.. However, there are a number of reasons for which one would prefer to fit together GR and Quantum Field Theory brill (); isham (). Let us list some of the most prominent ones here and leave the discussion about how to address them for the next subsection. Curiosity is probably the motivation leading scientific research. From this perspective it would be at least unusual if the gravity research community was so easily willing to abandon any attempt to describe the regime where both quantum and gravitational effects are important. The fact that the Planck scale seems currently experimentally inaccessible does not, in any way, imply that it is physically irrelevant. On the contrary, one can easily name some very important open issues of contemporary physics that are related to the Planck scale.
A particular example is the Big Bang scenario in which the Universe inevitably goes through an era in which its dimensions are smaller than the Planck scale (Planck era). On the other hand, space-time in GR is a continuum and so in principle all scales are relevant. From this perspective, in order to derive conclusions about the nature of space-time one has to answer the question of what happens on very small and very large scales.
i.3.1 UV scales: the Quantum Gravity Problem
One of the main challenges of modern physics is to construct a theory able to describe the fundamental interactions of nature as different aspects of the same theoretical construct. This goal has led, in the past decades, to the formulation of several unification schemes which, inter alia, attempt to describe gravity by putting it on the same footing as the other interactions. All these schemes try to describe the fundamental fields in terms of the conceptual apparatus of Quantum Mechanics. This is based on the fact that the states of a physical system are described by vectors in a Hilbert space and the physical fields are represented by linear operators defined on domains of . Until now, any attempt to incorporate gravity in this scheme has either failed or been unsatisfactory. The main conceptual problem is that the gravitational field describes simultaneously the gravitational degrees of freedom and the background space-time in which these degrees of freedom live.
Owing to the difficulties of building a complete theory unifying interactions and particles, during the last decades the two fundamental theories of modern physics, GR and Quantum Mechanics, have been critically re-analyzed. On the one hand, one assumes that the matter fields (bosons and fermions) come out from superstructures (e.g. Higgs bosons or superstrings) that, undergoing certain phase transitions, have generated the known particles. On the other hand, it is assumed that the geometry (e.g. the Ricci tensor or the Ricci scalar) interacts directly with quantum matter fields which back-react on it. This interaction necessarily modifies the standard gravitational theory, that is, the Lagrangian of gravity plus the effective fields is modified with respect to the Hilbert-Einstein one, and this fact can directly lead to the ETGs.
From the point of view of cosmology, the modifications of standard gravity provide inflationary scenarios of interest. In any case, a condition that must be satisfied in order for such theories to be physically acceptable is that GR is recovered in the low-energy limit.
Although remarkable conceptual progress has been made following the introduction of generalized gravitational theories, at the same time the mathematical difficulties have increased. The corrections introduced into the Lagrangian augment the (intrinsic) non-linearity of the Einstein equations, making them more difficult to study because differential equations of higher order than second are often obtained and because it is impossible to separate the geometric from the matter degrees of freedom. In order to overcome these difficulties and simplify the equations of motion, one often looks for symmetries of the Lagrangian and identifies conserved quantities which allow exact solutions of dynamics to be discovered. The key step in the implementation of this program consists of passing from the Lagrangian of the relevant fields to a point-like Lagrangian or, in other words, in going from a system with an infinite number of degrees of freedom to one with a finite number of degrees of freedom. Fortunately, this is feasible in cosmology because most models of physical interest are spatially homogeneous Bianchi models and the observed Universe is spatially homogeneous and isotropic to a high degree (Friedmann-Lemaitre-Robertson-Walker (FLRW) models).
The need for a quantum theory of gravity was recognized at the end of the 1950s, when physicist tried for the first time to treat all interactions at a fundamental level and to describe them in terms of Quantum Field Theory. Naturally, the first attempts to quantize gravity used the canonical approach and the covariant approach, which had been applied with remarkable success to Electromagnetism. In the first approach applied to Electromagnetism, one considers the electric and magnetic fields satisfying the Heisenberg Uncertainty Principle and the quantum states are gauge-invariant functionals generated by the vector potential defined on three-surfaces of constant time. In the second approach, one quantizes the two degrees of freedom of the Maxwell field without any 3+1 decomposition of the metric, while the quantum states are elements of a Fock space itzykson (). These procedures are equivalent in Electro-magnetism. The former allows for a well-defined measure, whereas the latter is more convenient for perturbative calculations such as, for example, the computation of the -matrix in Quantum Electro Dynamics (QED).
These methods have been applied also to GR, but many difficulties arise in this case due to the fact that Einstein’s theory cannot be formulated in terms of a quantum field theory on a fixed Minkowski background. To be more specific, in GR the geometry of the background space-time cannot be given a priori: space-time is the dynamical variable itself. In order to introduce the notions of causality, time, and evolution, one must first solve the equations of motion and then ”build” the space-time. For example, in order to know if particular initial conditions will give rise to a black hole, it is necessary to fully evolve them by solving the Einstein equations. Then, taking into account the causal structure of the obtained solution , one has to study the asymptotic metric at future null infinity in order to assess whether it is related, in the causal past, with those initial conditions. This problem becomes intractable at the quantum level. Due to the Uncertainty Principle, in non-relativistic Quantum Mechanics, particles do not move along well-defined trajectories and one can only calculate the probability amplitude that a measurement at time detects a particle around the spatial point . Similarly, in Quantum Gravity, the evolution of an initial state does not provide a specific space-time. In the absence of a space-time, how is it possible to introduce basic concepts such as causality, time, elements of the scattering matrix, or black holes?
The canonical and covariant approaches provide different answers to these questions. The canonical approach is based on the Hamiltonian formulation of GR and aims at using the canonical quantization procedure. The canonical commutation relations are the same that lead to the Uncertainty Principle; in fact, the commutation of certain operators on a spatial three-manifold of constant time is imposed, and this three-manifold is fixed in order to preserve the notion of causality. In the limit of asymptotically flat space-time, the motion generated by the Hamiltonian must be interpreted as temporal evolution (in other words, when the background becomes the Minkowski space-time, the Hamiltonian operator assumes again its role as the generator of translations). The canonical approach preserves the geometric features of GR without the need to introduce perturbative methods Wheeler57 (); ADM62 (); DeWitt68 (); Misner70 (); Misner72 ().
The covariant approach, instead, employs Quantum Field Theory concepts and methods. The basic idea is that the problems mentioned above can be easily circumvented by splitting the metric into a kinematical part (usually flat) and a dynamical part , as in
The geometry of the background is given by the flat metric tensor and is the same as in Special Relativity and ordinary Quantum Field Theory, which allows one to define the concepts of causality, time, and scattering. The quantization procedure is then applied to the dynamical field, considered as a (small) deviation of the metric from the Minkowski background metric. Quanta are discovered to be particles with spin two, called gravitons, which propagate in flat space-time and are defined by . Substituting the metric into the Hilbert-Einstein action, it follows that the Lagrangian of the gravitational sector contains a sum whose terms represent, at different orders of approximation, the interaction of gravitons and, eventually, terms describing matter-graviton interaction (if matter is present). Such terms are analyzed by using the standard techniques of perturbative Quantum Field Theory.
These quantization programs were both pursued during the 1960s and 1970s. In the canonical approach, Arnowitt, Deser, and Misner ADM62 () provided a Hamiltonian formulation of GR using methods proposed earlier by Dirac and Bergmann. In this Hamiltonian formalism, the canonical variables are the three-metric on the spatial submanifolds obtained by foliating the four-dimensional manifold (note that this foliation is arbitrary). The Einstein equations give constraints between the three-metrics and their conjugate momenta and the evolution equation for these fields, known as the Wheeler-DeWitt (WDW) equation. In this way, GR is interpreted as the dynamical theory of the three-geometries (geometrodynamics). The main difficulties arising from this approach are that the quantum equations involve products of operators defined at the same space-time point and, in addition, they entail the construction of distributions whose physical meaning is unclear. In any case, the main problem is the absence of a Hilbert space of states and, as consequence, a probabilistic interpretation of the quantities calculated is missing.
The covariant quantization approach is closer to the known physics of particles and fields in the sense that it has been possible to extend the perturbative methods of QED to gravitation. This has allowed the analysis of the mutual interaction between gravitons and of the matter-graviton interactions. The formulation of Feynman rules for gravitons and the demonstration that the theory might be unitary at every order of the expansion was achieved by DeWitt De Witt ().
Further progress was achieved with Yang-Mills theories, which describe the strong, weak, and electromagnetic interactions of quarks and leptons by means of symmetries. Such theories are renormalizable because it is possible to give the fermions a mass through the mechanism of Spontaneous Symmetry Breaking. Then, it is natural to attempt to consider gravitation as a Yang-Mills theory in the covariant perturbation approach and check whether it is renormalizable. However, gravity does not fit into this scheme; it turns out to be non-renormalizable when one considers the graviton-graviton interactions (two-loops diagrams) and graviton-matter interactions (one-loop diagrams).222Higher order terms in the perturbative series imply an infinite number of free parameters. At the one-loop level it is sufficient to renormalize only the effective constants and which, at low energy, reduce to Newton’s constant and the cosmological constant . The covariant method allows one to construct a theory of gravity which is renormalizable at one-loop in the perturbative series BirrellDavies (). Due to the non-renormalizability of gravity at different orders, its validity is restricted only to the low-energy domain, i.e., to large scales, while it fails at high energy and small scales. This implies that the full unknown theory of gravity has to be invoked near or at the Planck era and that, sufficiently far from the Planck scale, GR and its first loop corrections describe the gravitational interactions. In this context, it makes sense to add higher order terms to the Hilbert-Einstein action as we will do in the second part of this Report. Besides, if the free parameters are chosen appropriately, the theory has a better ultraviolet behavior and is asymptotically free. Nevertheless, the Hamiltonian of these theories is not bounded from below and they are unstable. In particular, unitarity is violated and probability is not conserved.
An alternative approach to the search for a theory of Quantum Gravity comes from the study of the Electroweak interaction. In this approach, gravity is treated neglecting the other fundamental interactions. The unification of the Electromagnetic and the weak interactions suggests that it might be possible to obtain a consistent theory when gravitation is coupled to some kind of matter. This is the basic idea of Supergravity van (). In this class of theories, the divergences due to the bosons (gravitons in this case) are cancelled exactly by those due to the fermions, leading to a renormalized theory of gravity. Unfortunately, this scheme works only at the two-loop level and for matter-gravity couplings. The Hamiltonian is positive-definite and the theory turns out to be unitary. But, including higher order loops, the infinities re-appear, destroying the renormalizability of the theory.
Perturbative methods are also used in String Theories. In this case, the approach is completely different from the previous one because the concept of particle is replaced by that of an extended object, the fundamental string. The usual physical particles, including the spin two graviton, correspond to excitations of the string. The theory has only one free parameter (the string tension) and the interaction couplings are determined uniquely. It follows that string theory contains all fundamental physics and it is therefore considered as a candidate for the Theory of Everything. String Theory seems to be unitary and the perturbative series converges implying finite terms. This property follows from the fact that strings are intrinsically extended objects, so that ultraviolet divergencies coming from small scales or from large transfer impulses, are naturally cured. In other words, the natural cutoff is given by the string length, which is of Planck size . At scales larger than , the effective string action can be rewritten in terms of non-massive vibrational modes, i.e., in terms of scalar and tensor fields (tree-level effective action). This eventually leads to an effective theory of gravity non-minimally coupled with scalar fields, the so-called dilaton fields.
To conclude, let us summarize the previous considerations:
a consistent (i.e., unitary and renormalizable) theory of gravity does not exist yet.
In the quantization program for gravity, two approaches have been used: the covariant approach and the perturbative approach. They do not lead to a definitive theory of Quantum Gravity.
In the low-energy regime (with respect to the Planck energy) at large scales, GR can be generalized by introducing into the Hilbert-Einstein action terms of higher order in the curvature invariants and non-minimal couplings between matter and gravity. These lead, at the one-loop level, to a consistent and renormalizable theory.
A part the lack of final theory, the Quantum Gravity Problem already contains some issues and shortcomings which could be already addresses by the today physics. We will summarize them in the forthcoming section.
i.3.2 Issues and shortcomings in Quantum Gravity
Considered the status of art, are some predictions of Quantum Gravity already available? Can remnants of Planck scale be detected at lower energy couplings and masses? As it is well known, only a fine-tuned combination of the low-energy constants leads to an observable Universe like ours. It would thus appear strange if a fundamental theory possessed just the right constants to achieve this. Hogan hogan () has argued that Grand Unified Theories constrain relations among parameters, but leave enough freedom for a selection. In particular, he suggests that one coupling constant and two light fermion masses are not fixed by the symmetries of the fundamental theory 333String theory contains only one fundamental dimensionfull parameter, the string length. The connection to low energies may nonetheless be non-unique due to the existence of many different possible ”vacua”... One could then determine this remaining free constants only by the (weak form of the) Anthropic Principle: they have values such that a Universe like ours is possible. The cosmological constant, for example, must not be much bigger than the presently observed value, because otherwise the Universe would expand too fast to allow the formation of galaxies. The Universe is, however, too special to be explainable on purely anthropic grounds. We know that the maximal entropy would be reached if all the matter in the observable Universe were collected into a single giant black hole. This entropy would be about , which is exceedingly more than the observed entropy of about . The ”probability” for our Universe would then be about . From the Anthropic Principle alone one would not need such a special Universe. As for the cosmological constant, for example, one could imagine its calculation from a fundamental theory. Taking the presently observed value for , one can construct a mass according to
which in elementary particle physics is not an unusally big or small value. The observed value of could thus emerge together with medium-size particle mass scales. Since fundamental theories are expected to contain only one dimensionfull parameter, low-energy constants emerge from fundamental quantum fields. An important example in string theory, is the dilaton field from which one can calculate the gravitational constant. In order that these fields mimic physical constants, two conditions have to be satisfied. First, decoherence must be effective in order to guarantee a classical behaviour of the field. Second, this ”classical” field must then be approximately constant in large-enough space-time regions, within the limits given by experimental data. The field may still vary over large times or large spatial regions and thus mimic a ”time- or space-varying constant”. The last word on any physical theory has to be spoken by experiments (observations). Apart from the possible determination of low-energy constants and their dependence on space and time, what could be the main tests for Quantum Gravity?
Black-hole evaporation: A key test would be the final evaporation phase of a black hole. To this end, it would be useful to observe signatures of primordial black holes. These objects are forming not at the end of stellar collapse, but they can result from strong density perturbations in the early Universe. In the context of inflation, their initial mass can be as small as 1 g. Primordial black holes with initial masses of about g would evaporate at the present age of the Universe. Unfortunately, no such object has yet been observed. Especially promising may be models of inflationary cosmology acting at different scales Bringmann ().
Cosmology: Quantum aspects of gravitational field may be observed in the anisotropy spectrum of the cosmic microwave background. First, future experiments may be able to observe the contribution of the gravitons generated in the early Universe. This important effect was already emphasized in staro (). The production of gravitons by the cosmological evolution would be an effect of Quantum Gravity. Second, quantum-gravitational correction terms from the Wheeler-DeWitt equation or its generalization in loop quantum cosmology may leave their impact on the anisotropy spectrum. Third, a discreteness in the inflationary perturbations could manifest itself in the spectrum hogan ().
Discreteness of of space and time: Both in String Theory and Quantum Gravity there are hints of a discrete structure of space-time. This quantum foam could be seen through the observation of effects violating local Lorentz invariance amelino (), for example, in the dispersion relation of the electromagnetic waves coming from gamma-ray bursts. It has even be suggested that space-time fluctuations could be seen in atomic interferometry Percival (). However, there exist severe observational constraints Peters ().
Signatures of higher dimensions: An important feature of String Theory is the existence of additional space-time dimensions. They could manifest themselves in scattering experiments at the Large Hadron Collider (LHC) at CERN. It is also imaginable that they cause observable deviations from the standard cosmological scenario LHC ().
Some of these features are discussed in detail in Kimberly (). Of course, there may be other possibilities which are not yet known and which could offer great surprises. It is, for example, imaginable that a fundamental theory of Quantum Gravity is intrinsically non-linear Penrose (); Singh (). This is in contrast to most currently studied theories of Quantum Gravity, which are linear. Quantum Gravity has been studied since the end of the 1920s. No doubt, much progress has been made since then. The final goal has not yet been reached. The belief expressed here is that a consistent and experimentally successful theory of Quantum Gravity will be available in the future. However, it may still take a while before this time is reached. In any case, ETGs could constitute a serious approach in this direction.
i.3.3 IR scales: Dark Energy and Dark Matter
The revision of standard early cosmological scenarios leading to inflation implies that a new approach is necessary also at later epochs: ETGs could play a fundamental role also in this context. In fact, the increasing bulk of data accumulated over the past few years has paved the way for a new cosmological model usually referred to as the Concordance Model or Cold Dark Matter (CDM) model.
The Hubble diagram of type Ia supernovae (hereafter SNeIa) measured by both the Supernova Cosmology Project Knopetal03 (); Perlmutteretal03 () and the High- Team Riessetal98 (); Tonryetal03 () up to redshift , was the first piece of evidence that the Universe is currently undergoing a phase of accelerated expansion. Besides, balloon-born experiments such as BOOMERANG deBernardisetal00 () and MAXIMA Maxima () determined the location of the first two Doppler peaks in the spectrum of Cosmic Microwave Background (CMB) anisotropies, strongly suggesting a Universe with flat spatial sections. When combined with the constraints on the matter density parameter , these data indicate that the Universe is dominated by an unclustered fluid with negative pressure commonly referred to as dark energy, which drives the accelerated expansion. This picture has been further strengthened by the recent precise measurements of the CMB spectrum by the WMAP satellite experiment WMAP (); hinshaw (); Spergel:2006hy (), and by the extension of the SNeIa Hubble diagram to redshifts larger than one Riess04 ().
An overwhelming number of papers appeared following these observational evidences, which present a large variety of models attempting to explain the cosmic acceleration. The simplest explanation would be the well known cosmological constant SahniStarobinsky00 (). Although the latter provides the best-fit to most of the available astrophysical data WMAP (), the CDM model fails egregiously in explaining why the inferred value of is so tiny (120 orders of magnitude lower) in comparison with the typical value of the vacuum energy density predicted by particle physics, and why its present value is comparable to the matter density, this is the so-called coincidence problem.
As a tentative solution, many authors have replaced the cosmological constant with a scalar field rolling slowly down a flat section of a potential and giving rise to the models known as quintessence QuintRev (); tsu1 (). Albeit successful in fitting the data with many models, the quintessence approach to dark energy is still plagued by the coincidence problem since the dark energy and dark matter densities evolve differently and reach comparable values only during a very short time of the history of the Universe, coinciding in order of magnitude right at the present era. In other words, the quintessence dark energy is tracking matter and evolves in the same way for a long time; at late times, somehow it changes its behaviouor and no longer tracks the dark matter but begins to dominate in the fashion of a (dynamical) cosmological constant. This is the coincidence problem of quintessence.
Furthermore, the origin of this quintessence scalar field is mysterious, although various (usually rather exotic) models have been proposed, leaving a great deal of uncertainty on the choice of the scalar field potential necessary to achieve the late-time acceleration of the Universe. The subtle and elusive nature of dark energy has led many authors to look for a completely different explanation of the acceleration behaviour of the cosmos without introducing exotic components. To this end, it is worth stressing that the present-day cosmic acceleration only requires a negative pressure component that comes to dominate the dynamics late in the matter era, but does not tell us anything about the nature and the number of the cosmic fluids advocated to fill the Universe. This consideration suggests that it could be possible to explain the accelerated expansion with a single cosmic fluid characterized by an equation of state causing it to act like dark matter at high densities, while behaving as dark energy at low densities. An attractive feature of these models, usually referred to as Unified Dark Energy (UDE) or Unified Dark Matter (UDM) models, is that the coincidence problem is solved naturally, at least at the phenomenological level, with such an approach odi2006 (); odi20062 (). Examples are the generalized Chaplygin gas Chaplygin (), the tachyon field tachyon (), and condensate cosmology Bassett (). A different class of UDE models with a single fluid has been proposed Hobbit1 (); Hobbit2 (): its energy density scales with the redshift in such a way that a radiation-dominated era, followed by a matter era and then by an accelerating phase can be naturally achieved. These models are extremely versatile since they can be interpreted both in the framework of UDE or as two-fluid scenarios containing dark matter and scalar field dark energy. A characteristic feature of this approach is that a generalized equation of state can always be obtained and the fit to the observational data can be attempted. However, such models explain the phenomenology but cannot be addressed to some fundamental physics.
There is another, different, way to approach the problem of the cosmic acceleration. As stressed in LSS03 (), it is possible that the observed acceleration is not the manifestation of yet another ingredient of the cosmic pie, but rather the first signal of a breakdown, in the infrared limit, of the laws of gravitation as we understand them. From this point of view, it is tempting to modify the Einstein-Friedmann equations to see whether it is still possible to fit the astrophysical data with models containing only standard matter without exotic fluids. Examples are the Cardassian expansion Cardassian () and Dvali-Gabadadze-Porrati (DGP) gravity DGP (). Within the same conceptual framework, it is possible to find alternative schemes in which a quintessential behaviour is obtained by incorporating effective models coming from fundamental physics and giving rise to generalized or higher order gravity actions Capozziello02IJMPD () (see Ref. CF1 (); booksalv (); defelice (); bookfelix (). For instance, a cosmological constant may be recovered as a consequence of a non-vanishing torsion field, leading to a model consistent with both the SNeIa Hubble diagram and observations of the Sunyaev–Zel’dovich effect in galaxy clusters torsion (). SNeIa data could also be efficiently fitted by including in the gravitational sector higher order curvature invariants CapozzielloCardoneCarloniTroisi03 (); Li:2007xn (); CF1 (); Li:2007xw (). These alternative models provide naturally a cosmological component with negative pressure originating in the geometry of the Universe, thus overcoming the problematic nature of quintessence scalar fields. Cosmological models coming from ETGs are in the track of this philosophy.
The variety of cosmological models which are phenomenologically viable candidates to explain the observed accelerated expansion is clear from this short review. This overabundance signals that only a limited number of cosmological tests are available to discriminate between competing theories, and it is clear that there is a high degeneracy of models. Let us remark that both the SNeIa Hubble diagram and the angular size-redshift relation of compact radio sources AngTest2 () are distance-based probes of the cosmological model and, therefore, systematic errors and biases could be iterated. With this point in mind, it is interesting to search for tests based on time-dependent observables. For example, one can take into account the lookback time to distant objects since this quantity can discriminate between different cosmological models. The lookback time is observationally estimated as the difference between the present-day age of the Universe and the age of a given object at redshift . This estimate is possible if the object is a galaxy observed in more than one photometric band since its colour is determined by its age as a consequence of stellar evolution. Hence, it is possible to obtain an estimate of the galaxy’s age by measuring its magnitude in different bands and then using stellar evolutionary codes to choose the model that best reproduces the observed colours andreon (); ester ().
Coming to the weak-field limit, which essentially means considering Solar System scales, any alternative relativistic theory of gravity is expected to reproduce GR which, in any case, is firmly tested only in this limit and at these scales Will93 (). Even this limit is a matter of debate since several relativistic theories do not reproduce exactly the Einsteinian results in their Newtonian limit but, in some sense, generalize them. As was first noticed by Stelle Stelle:1976gc (), -gravity gives rise to Yukawa-like corrections to the Newtonian potential with potentially interesting physical consequences. For example, it is claimed by some authors that the flat rotation curves of galaxies can be explained by such terms Sanders90 (). Others mannheim () have shown that a conformal theory of gravity is nothing else but a fourth order theory containing such terms in the Newtonian limit. Reports of an apparent anomalous long-range acceleration in the data of the Pioneer 10/11, Galileo, and Ulysses spacecrafts could be accommodated in a general theoretical scheme incorporating Yukawa corrections to the Newtonian potential Anderson02 (); bertolami ().
In general, any relativistic theory of gravitation yields corrections to the weak-field gravitational potentials (e.g., Qua91 ()) which, at the post-Newtonian level and in the Parametrized Post-Newtonian (PPN) formalism, could constitute a test of these theories Will93 (). Furthermore, the newborn gravitational lensing astronomy ehlers () is providing additional tests of gravity over small, large, and very large scales which will soon provide direct measurements of the variation of the Newton coupling krauss (), the potential of galaxies, clusters of galaxies, and several other features of self-gravitating systems. Very likely, such data will be capable of confirming or ruling out as GR or ETGs.
This short overview shows that several shortcomings point out that GR cannot to be the final theory of gravity notwithstanding its successes in addressing a large amount of theoretical and experimental issues. ETGs could be a viable approach to solve some of these problems at IR and UV scales without pretending to be the comprehensive and self-consistent fundamental theory of gravity but in the track outlined by GR and then in the range of gauge theories. This review paper is mainly devoted to the theoretical foundation of ETGs trying to insert them in the framework of gauge theories and showing that they are nothing else but a straightforward extension of GR. The cosmological phenomenology and the genuinely astrophysical aspects of ETGs are not faced here. We refer the readers to the excellent reviews and books quoted in the bibliography mybook (); booksalv (); mauro (); defelice (); CF1 (); lucaamendola (); tsu1 (); salzano (); faraoni-noi (); lobo (); reportodi (); cliftonrep ().
Ii A summary of gauge symmetries
Modern gauge theory has emerged as one of the most significant developments of physics of XX century. It has allowed us to realize partially the issue of unifying the fundamental interactions of nature. We now believe that the electromagnetism, which has been long studied, has been successfully unified with the nuclear weak interaction, the force to which radioactive decay is due. What is the most remarkable about this unification is that these two forces differ in strength by a factor of nearly . This important accomplishment by the Weinberg-Salam gauge theory weinberg (), and insight gained from it, have encouraged the hope that also the other fundamental forces could be unified within a gauge theory framework. At the same time, it has been realized that the potential areas of application for gauge theory extended far beyond elementary particle physics. Although much of the impetus for a gauge theory came from new discoveries in particle physics, the basic ideas behind gauge symmetry have also appeared in other areas as seemingly unrelated, such as condensed matter physics, non-linear wave phenomena and even pure mathematics. This great interest in gauge theory indicates that it is in fact a very general area of study and not only limited to elementary particles.
Gauge invariance was recognized only recently as the physical principle governing the fundamental forces between the elementary particles. Yet the idea of gauge invariance was first proposed by Hermann Weyl in 1919 when only the electron and proton weyl () were known as fundamental particles. It required nearly 50 years for gauge invariance to be ”rediscovered” and its significance to be understood. The reason for this long hiatus was that Weyl’s physical interpretation of gauge invariance was shown to be incorrect soon after he had proposed the theory. Gauge invariance only managed to survive because it was known to be a symmetry of Maxwell’s equations and thus became a useful mathematical help in order to simplify equations and thus became a useful mathematical device for simplifying many calculations in the electrodynamics. In view of present success of gauge theory, we can say that gauge invariance was the classical case of a good idea which was discovered long before its time.
In this section, we present a brief summary of gauge theory in view of the fact that any theory of gravity can be considered under the same standard. The early history of gauge theory can be divided into old and new periods where the dividing can be set in the 1950’s. The most important question is what motivated Weyl to propose the idea of gauge invariance as a physical symmetry? How did he manage to express it in a mathematical form that has remained almost the same today although the physical interpretation has altered radically? And, how did the development of Quantum Mechanics lead Weyl himself to a rebirth of a gauge theory? The new period of gauge theory begins with the pioneering attempt of Yang and Mills to extend gauge symmetry beyond the narrow limits of electromagnetism Yang-Mills (). Here we will review the radically new interpretation of gauge invariance required by Yang-Mills theory and the reasons for the failure of original theory. By comparing the new theory with that of Weyl, we can see that many original ideas of Weyl have been rediscovered and incorporated into the modern theory moriyasu ().
In these next subsection, our purpose is to present an elementary introduction to a gauge theory in order to show that any relativistic theory of gravity is a gauge theory.
ii.1 What is a gauge symmetry?
In physics, gauge invariance (also called gauge symmetry) is the property of a field theory where different configurations of the underlying fundamental but unobservable fields result in identical observable quantities. A theory with such a property is called a gauge theory. A transformation from a field configuration to another is called a gauge transformation. Modern physical theories describe nature in terms of fields, e.g., the electromagnetic field, the gravitational field, and fields for the electron and all other elementary particles. A general feature of these theories is that none of these fundamental fields, which are the fields that change under a gauge transformation, can be directly measured. On the other hand, the observable quantities, namely the ones that can be measured experimentally as charges, energies, velocities, etc. do not change under a gauge transformation, even though they are derived from the fields that do change. This (and any) kind of invariance under a transformation is called a symmetry.
For example, in classical electromagnetism the Electric field, , and the magnetic field, , are observable, while the underlying and more fundamental electromagnetic potentials and are not. Under a gauge transformation which jointly alters the two potentials, no change occurs either in or or in the motion of charged particles. In this example, the gauge transformation was just a mathematical feature without any physical relevance, except that gauge invariance is intrinsically connected to the fundamental law of charge conservation. As shown above, with the advent of Quantum Mechanics in the 1920s, and with successive Quantum Field Theory, the importance of gauge transformations has steadily grown. Gauge theories constrain the laws of physics, because of the fact that all the changes induced by a gauge transformation have to cancel each other out when written in terms of observable quantities. Over the course of the 20th century, physicists gradually realized that all forces (fundamental interactions) arise from the constraints imposed by local gauge symmetries, in which case the transformations vary from point to point in space and time. Perturbative quantum field theory (usually employed for scattering theory) describes forces in terms of force mediating particles called gauge bosons. The nature of these particles is determined by the nature of the gauge transformations. The culmination of these efforts is the Standard Model, a quantum field theory explaining all of the fundamental interactions except gravity.
ii.2 The Einstein connection
In 1919, people thought that only two fundamental forces of nature existed, Electromagnetism and Gravitation. In that same year, a group of scientists also made the first experimental observation of starlight bending in the gravitational field of the sun during a total eclipse kluber (). The brilliant confirmation of Einstein’s General Theory of Relativity inspired Weyl to propose his own revolutionary idea of gauge invariance in 1919. To see how this came about, let us first briefly recall some basic ideas on which Relativity was built. The fundamental concept underlying both Special and GR is that are no absolute frames of reference in the Universe. The physical motion of any system must be described relatively to some arbitrary coordinate frame specified by an observer, and the laws of physics must be independent of the choice of frame. In Special Relativity, one usually, defines convenient reference frames, which are called ”inertial”, in motion with uniform velocity. For example, consider a particle which is moving with constant velocity with respect to an observer. Let be the rest frame of the observer and be an inertial frame which is moving at the same velocity as the particle. The observer can either state that the particle is moving with velocity in or that it is at rest . The important point to be noted from this trivial example is that the inertial frame can always be related by a simple Lorentz transformation to the observer frame . The transformation depends only on the relative velocity between the and observers, not on their positions in space-time. The particle and observer can be infinitesimally close together or at opposite ends of the Universe; the Lorentz transformation is still the same. Thus the Lorentz transformation, or rather the Lorentz symmetry group of Special Relativity, is an example of ”global” symmetry. In GR, the description of relative motion is much more complicated because now one is dealing with the motion of a system embedded in a gravitational field. As an illustration, let us consider the following ”gedanken” experiment for measuring the motion of a test particle which is moving through a gravitational field. The measurement is to be performed by a physicist in an elevator. The elevator cable as broken so that the elevator and the physicist are falling freely bergmann (). As the particle moves through the field, the physicist determines its motion with respect to the elevator. Since both particle and elevator are falling in the same field, the physicist can describe the particle motion as if there were no gravitational field. The acceleration of the elevator cancels out the acceleration of particle due to gravity. This example of the Principle of Equivalence, follows from the well-known fact that all bodies accelerate at the same rate given the gravitational field (e.g. on the surface of the Earth). Let us now compare the physicist in the falling elevator with the observer in the inertial frame in Special Relativity. It could appear that the elevator corresponds to an accelerating or ”non-inertial” frame that is analogous to the frame in which the particle appeared to be at rest. However, it is not true that a real gravitational field does not produce the same acceleration at every point in space. As one moves infinitely far away from the source, the gravitational field will eventually vanish. Thus, the falling elevator can only be used to define a reference frame within an infinitesimally small region where the gravitational field can be considered uniform. Over a finite region, the variation of the field may be sufficiently large for the acceleration of the particle not to be completely cancelled. We see that an important difference between Special Relativity and GR is that a reference frame can only be defined ”locally” or at a single point in a gravitational field. This creates a fundamental problem. To illustrate this difficult point, let us now suppose that there are many more physicists in nearby falling elevators. Each physicist makes an independent measurement so that the path of particle in the gravitational field can be determined. The measurements are made in separate elevators at different locations in the field. Clearly, one cannot perform an ordinary Lorentz transformation between the elevators. If the different elevators were related only by Lorentz transformation, the acceleration would have to be independent of position and the gravitational field could not decrease with distance from the source. Einstein solved the problem of relating nearby falling frames by defining a new mathematical relation known as ”connection”. To understand the meaning of a connection, let us consider a -vector which represents some physically measured quantity. Now suppose that the physicist in the elevator located at observes that changes by an amount and a second physicist in a different elevator at observes a change in . In Special Relativity, the differential is also a vector like itself. Thus, the differential in the elevator at is given by the relation 444The components of the -vector and with . Vector components with upper and lower indices are related by , where is the metric tensor which appears in the definition of the invariant space-time interval . The components of are , and all other components are zero.
where . The relation (2) follows from the fact that the Lorentz transformation between and is a linear transformation. We can no longer assume that the transformation from to is linear in GR. Thus, we must write for the general expression
Clearly, the second derivatives will vanish if the are linear functions of the . Such terms are actually quite familiar in physics. They occur in ”curvilinear” coordinate systems. These curvilinear coefficients are denoted by the symbol
and are called the components of a ”connection”. They are also called affine connections or Christoffel symbols gravitation (). It is important to note that the gravitational connection is not simply the result of using a curvilinear coordinate system. The value of the connection at each point in space-time is dependent on the properties of the gravitational field. The field is important in the determination of the relative orientation of the different falling elevators in the same way that the ”upward” direction on the surface of the earth varies from one position to another. The analogy with curvilinear coordinate systems merely indicates that the mathematical descriptions of free-falling frames and curvilinear coordinates are similar. Einstein generalized this similarity and arrived at the revolutionary idea of replacing gravity by the curvature of space-time einstein-gauge (). Let us briefly summarize the essential characteristics of GR that Weyl would have utilized for his new gauge theory. First of all, GR involves a specific force, gravitation, which is not studied in Special Relativity. However, by studying the properties of coordinates frames just as in Special Relativity, one learns that only local coordinates can be defined in a gravitational field. This local property is required by the physical behavior of the field and leads naturally to the idea of a connection between local coordinate frames. Thus the essential difference between Special Relativity and GR is that the former is a global theory while the latter is a local theory. This local property was the key to Weyl’s gauge theory moriyasu (). In Sec. III we will develope extensively this idea.
ii.3 The Weyl Gauge
Weyl went a step beyond GR and asked the following question: if the effects of a gravitational field can be described by a connection which gives the relative orientation between local frame in space-time, can other forces of nature such as Electromagnetism also be associated with similar connections? Generalizing the idea that all physical measurements are relative, Weyl proposed that absolute magnitude or norm of a physical vector also should not be an absolute quantity but should depend on its location in space-time. A new connection would then be necessary in order to relate the lengths of vectors at different positions. This is the scale or ”gauge” invariance. It is important to note here that the true significance of Weyl’s proposal lies on the local property of gauge symmetry and not in a special choice of the norm or ”gauge” as a physical variable. The assumption of locality is a powerful condition that determines not only the general structure but many of the detailed features of gauge theory. Weyl’s gauge invariance can be easily expressed in mathematical form yang ().Let us consider a vector at position with norm given by . If we shift the vector or transform the coordinates so that the vector is now at , the norm becomes . Expanding to first order in , we can write the norm as
We now introduce a gauge change through a multiplicative scaling factor . The factor is defined for convenience to equal unity at the position . The scale factor at is then given by
For a constant vector, we see that the norm has changed by an amount
Te derivative is the new mathematical ”connection” associated with the gauge change. Weyl identified the gauge connection with the electromagnetic potential . It is straightforward to show that a second gauge change with a scale factor will transform the connection as follows,
From classical Electromagnetism, we known that the potential behaves under gauge transformation like
which leaves the electric and magnetic fields unchanged. Since the forms of (9) and (10) are identical, it appeared that Weyl’s new interpretation of the potential as a gauge connection was perfectly compatible with Electromagnetism. Unfortunately, it was soon pointed out that the basic idea of scale invariance itself would lead to conflict with known physical facts bergmann1 (). Some years later, Bergmann noted, that Weyl’s original interpretation of gauge invariance would also be in conflict with Quantum Theory. The wave description of matter defines a natural scale for a particle through its Compton wavelength . Since the wavelength is determined by the particle mass , it cannot depend on position and thus contradicts Weyl’s original assumption about scale invariance. Despite the initial failure of Weyl’s gauge theory, the idea of a local gauge symmetry survived. It was well known that Maxwell’s equations were invariant under a gauge change. However, without an acceptable interpretation, gauge invariance was regarded as only an ”accidental” symmetry of Electromagnetism. The gauge transformation property in Eq. (10) was interpreted as just a statement of the well known arbitrariness of the potential in classical physics. Only the electric and magnetic fields were considered to be real and observable. Gauge symmetry was retained largely because it was useful for calculations in both classical and quantum electrodynamics. In fact a lot of problems in electrodynamics can often be most easily solved by first choosing a suitable gauge, such as the Coulomb gauge or Lorentz gauge, in order to make the equations more tractable moriyasu ().
ii.4 Electromagnetism as a Gauge Theory
It is clear that the electromagnetic interaction of charged particles could be interpreted as a local gauge theory within the context of Quantum Mechanics. In analogy with Weyl’s first theory, the phase of a particle wavefunction can be identified as a new physical degree of freedom which is dependent on the space-time position. The value can be changed arbitrarily by performing purely mathematical phase transitions on the wavefunction at each point. Therefore, there must be some connections between phase value nearby points. The role of this connection is payed by the electromagnetic potential. This strict relation between potential and the change in phase is clearly demonstrated by the Aharonov–Bohm effect aharonov (). Thus by using the phase of wavefunction as the local variable instead of the norm of a vector, Electromagnetism can be interpreted as a local gauge theory very much as Weyl envisioned. Gauge transformations can be viewed as merely phase changes so that they look more like a property of Quantum Mechanics than Electromagnetism. In addition, the symmetry defined by the gauge transformations does not appear to be ”natural”. The set of all gauge transformations forms a one-dimensional unitary group known as the group. This group does not arise from any form of coordinate transformation like the more familiar spin-rotation group or Lorentz group. Thus, one has lost the original interpretation proposed by Weyl of a new space-time symmetry. The status of gauge theory was also influenced by the historical fact that Maxwell had formulated Electromagnetism long before Weyl proposed the idea of gauge invariance. Therefore, unlike the GR, the gauge symmetry group did not play any essential role in defining the dynamical content of Electromagnetism. This sequence of events was to be completly reversed in the development of modern gauge theory jackson (); goldstein (); moriyasu ().
ii.5 The Yang-Mills Gauge
In 1954, C. N. Yang and R. Mills proposed that the strong nuclear interaction can be described by a field theory like Electromagnetism. They postulated that the local gauge group was the isotopic-spin group. This idea was revolutionary because it changed the very concept of the ”identity” of an elementary particle. If the nuclear interaction is a local gauge theory like Electromagnetism, then there is a potential conflict with the notion of how a particle state. For example, let us assume that we can ”turn off” the electromagnetic interaction so that we cannot distinguish the proton and neutron by electric charge. We also ignore the small mass difference. We must then label the proton as the ”up” state of isotopic spin and the neutron as the ”down” state. But if isotopic spin invariance is an independent symmetry at each point in space-time, we cannot assume that the ”up” state is the same at any other point. The local isotopic spin symmetry allows to choose arbitrarily which direction is ”up” at each point without reference to any other point. Given that the labelling of a proton or a neutron is arbitrary at each point, once the choice has been made at one location, it is clear that some rule is then needed in order to make a comparison with the choice at any other position. The required rule, as Weyl proposed originally, is supplied by a connection. A new isotopic spin potential field was therefore postulated by Yang and Mills in analogy with the electromagnetic potential. However, the greater complexity of the isotopic-spin group as compared to the phase group means that Yang-Mills potential will be quite different from the electromagnetic field. In Electromagnetism, the potential provides a connection between the phase values of the wavefunction at different positions. In the Yang-Mills theory, the phase is replaced by a more complicated local variable that specifies the direction of the isotopic spin. In order to understand qualitatively how this leads to a connection, we need only to recall that the isotopic-spin group is also the group of rotations in a 3-dimensional space 555Technically, the group is different from the group of 3-dimensional rotations, ; the group is the ”covering group” of .. As an example, let us visualize the ”up” component of isotopic spin as a vector in an abstract ”isotopic spin space”. An obvious way to relate the ”up” states at different locations and is to ask how much the ”up” state at needs to be rotated so that it is oriented in the same direction as the ”up” state at . This suggests that the connection between isotopic spin states at different points must act like isotopic spin rotation itself. In other words, if a test particle in the ”up” state at is moved through the potential field to position , its isotopic spin direction must be rotated by the field so that it is pointing in the ”up” direction corresponding to the position . We can immediately generalize this idea to states of arbitrary isotopic spin. Since the components of an isotopic spin state can be transformed into another one by elements of the group, we can conclude that the connection must be capable of performing the same isotopic spin transformations as the group itself. This idea that the isotopic spin connection, and therefore the potential, acts like the symmetry group is the most important result of the Yang–Mills theory. This concept lies at the heart of the local gauge theory. It shows explicitly how the gauge symmetry group is built into the dynamics of the interaction between particles and fields. How is ti possible for a potential to generate a rotation in an internal symmetry space? To answer this question, we must define the Yang–Mills potential more carefully in the language of the rotation group. A 3-dimentional rotation of a wavefunction is written as
where is the angle of rotation and is the angular momentum operator. Let us compare this rotation with the phase change of wavefunction after a gauge transformation. The rotation has the same mathematical form as the phase factor of the wavefunction. However, this does not mean that the potential itself is a rotation operator like . We noted earlier that the amount of phase change must also be proportional to the potential in order to ensure that Schrödinger equation remains gauge invariant. To satisfy this condition, the potential must be proportional to the angular momentum operator in (11). Thus, the most general form of the Yang–Mills potential is a linear combination of the angular momentum operators
where the coefficients depend on the space-time position and we explicitly write the sum over the components. This relation indicates that the Yang–Mills is not a rotation, but rather is a ”generator” of a rotation. For the case of Electromagnetism, the angular momentum operator is replaced by a unit matrix and is just proportional to the phase change . The relation (12) explicitly displays the dual role of the Yang–Mills potential as both a field in space-time and an operator in the isotopic-spin space.
We can immediately deduce some interesting properties of the Yang–Mills potential. For example, the potential must have three charge components corresponding to the three independent angular momentum operators , and . The potential component which acts like a raising operator can transform a ”down” state into a ”up” state. We can associate this formal operation with a real process where a neutron absorbs a unit of isotopic spin from the gauge field and turns into a proton. This example indicates that the Yang–Mills gauge field must itself carry electric charge unlike electromagnetic potential. The Yang–Mills field also differs in other respects from the electromagnetic field but they both have one property in common, namely, they have zero mass. The zero mass of the photon is well known from Maxwell’s equations, but local gauge invariance requires that the mass of the gauge potential field be identically zero for any gauge theory. The reason is that the mass of the potential must be introduced into a Lagrangian through a term of the form
This guarantees that the correct equation of motion for a vector field will be obtained from the Euler-Lagrange equations. Unfortunately, the term given by (13) is not invariant under a gauge transformation. The special transformation property of the potential will introduce extra terms in (13) proportional to , which are not cancelled by the transformation of the wavefunction. Thus, the standard mass term is not allowed in the Yang–Mills gauge field must have exactly zero mass like the photon. The Yang–Mills field will therefore exhibit long-range behaviour like Coulomb field and cannot reproduce the observed short range of the nuclear force. Since this conclusion appeared to be an inescapable consequence of a local gauge invariance, the Yang–Mills theory was not considered to be an improvement on the already existing theories of the strong nuclear interaction.
Although the Yang–Mills theory field in its original purpose, it established the foundation for modern gauge theory. The isotopic-spin gauge transformation could not be regarded as a mere phase change; it required an entirely new interpretation of a gauge invariance. Yang and Mills showed for the first time that local gauge symmetry was a powerful fundamental principle that could provide new insight into the newly discovered ”internal” quantum numbers like isotopic spin. In the Yang–Mills theory, isotopic spin was not just a label for the charge states of particles, but it was crucially involved in determining the fundamental form of the interaction moriyasu ().
ii.6 Geometry and Gauge
The Yang–Mills theory revived the old ideas that elementary particles have degrees of freedom in some ”internal” space. By showing how these internal degrees of freedom could be unified in a non-trivial way with the dynamical motion in space-time, Yang and Mills discovered a new type of geometry in physics.
The geometrical structure of a gauge theory can be seen by comparing the Yang–Mills theory with of GR. The essential role of the connection is evident in both gauge theory and relativity. There is an analogy between non-inertial coordinate frame and gauge theory but the local frame has to be is located in an abstract space associated with the gauge symmetry group. To see how the gauge group defines an internal space, let us examine the examples of the phase group and the isotopic spin group. For the group, the internal space consists of all possible values of the phase of the wavefunction. These phase values can be interpreted as angular coordinates in a 2-dimensional space. The internal symmetry space of thus looks like a ring, and the coordinate of each point in this space is just the phase value itself. The internal space defined by group is more complicated because it describes rotation in a 3-dimensional space.
We recall that the orientations of an isotopic spin state can be generated by starting from a fixed initial isotopic direction, which can be chosen as the isotopic spin ”up” direction, and then rotating to the desired final direction. The values of the three angles which specify the rotation can be considered as the coordinates for a point inside an abstract 3-dimensional space. Each point corresponds to a distinct rotation so that the isotopic spin states themselves can be identified with the points in this angular space. Thus, the internal symmetry space of the group looks like the interior of a 3-dimensional sphere.
The symmetry space of a gauge group provides the local non-inertial coordinate frame for the internal degrees of freedom. To an imaginary observer inside this internal space, the interaction between a particle and an external gauge field looks like a rotation of the local coordinates. The amount of the rotation is determined by the strength of the external potential, and the relative change in the internal coordinates between two space-time points is just given by the connection as stated before. Thus, we see that there is a similarity between the geometrical description of relativity and the internal space picture.
The internal space is called a ”fiber” by mathematicians choque (); dreschsler (). The idea of using a gauge potential to ”link” together space-time with internal symmetries space is a new concept in physics. The new space formed by the union of 4-dimensional space time with an internal space is called a ”fiber bundle” space. The reason for this name is that the internal spaces or ”fibers” at each space-time point can all be viewed as the same space because they can be transformed into each other by a gauge transformation. Hence, the total space is a collection or ”bundle” of fibers.
Given that Yang and Mills developed their theory using the same terminology as electrodynamics, it is relevant to ask if there are good reasons to describe gauge theory in geometrical terms, other than to establish a historical link with relativity. The best reason for doing so is that the geometrical picture provides a valuable aid to the standard language of field theory. Most of the pedagogical aids in field theory are based on a long familiarity with electrodynamics. Modern gauge theory, on the other hand, requires a new approach in order to deal with all a of the fundamental forces between elementary particles. The geometrical picture can provide a common arena for discussing electromagnetism, the electromagnetism, the strong and weak nuclear forces, and even gravity, because it depends on only very general properties of gauge theory. The fiber bundle representation will be reconsidered in Sec. XII to discuss ETGs.
ii.7 Local gauge transformations
We have described qualitatively how the gauge group is associated with a connection. Any particle or system which is localized in a small volume and carries an internal quantum number like isotopic spin has a direction in the internal symmetry space. This internal direction can be arbitrarily chosen at each point in space-time. In order to compare these internal space directions at two different space-time points and , we need to define an appropriate connection which can tell us how much the internal direction at differs from the direction . This connection must be capable of relating all possible directions in the internal space to each other. The most obvious way to relate two directions is to find out how much one direction has to rotate so that it agrees with the other direction. The set of all such rotations forms a symmetry group; thus, the connection between inertial space directions at different points act like a symmetry group as well. Our problem now is to see how a symmetry group transformation can lead to a connection which we identify with a gauge potential field. Let us, begin by writing the general form of a local symmetry transformation for an arbitrary (non Abelian) group,
The ”local” nature of the transformation is indicated by the parameters which are continuous of . The constant is the electric charge for the gauge group or a general ”coupling constant” for an arbitrary gauge group. This is the only way in which the charge enters directly into the calculation. The general transformation (14) is identical to the usual form of an ordinary spatial rotation if we identify the position-dependent parameter with rotation angles. The are the generators of the internal symmetry group and satisfy the usual commutation relations
where the structure constant depend on the particular group. For the isotopic-spin rotation group , the generators, are the angular momentum operators. To see how the transformation (14) defines a connection, let us consider the following simple operation. We will take a test particle described by a wavefunction and move it between two points and in space-time, and analyze how its direction changes in the internal symmetry space. The internal direction at is initially chosen to have the angles . As the test particle moves away form , the internal direction changes in some continuous way until it reaches where it has new internal direction given angles . For an infinitesimal distance , this change can be described by the transformation (14) acting on and producing a rotation of the internal direction equal to the difference . This rotation gives us what we need, namely, a connection between internal space directions at different points in space-time. We also see that this connection involves the derivative of a quantity just like the connection defined by Weyl. In this case, the quantities are the internal rotation angles . This is a straightforward generalization of the phase of a wavefunction to a set of angles which specify the internal direction.
ii.8 Connections and potentials
Let us see how to calculate the connection from the symmetry transformation (14) by moving the charged test particle through an external potential field. We will explicitly separate the particle wavefunction into external and internal parts. Let us write
where form a set o a ”basis vectors” in the internal space. The index is an internal label such as the components of isotopic spin. The basis is analogous to the local non-inertial frame in relativity. The external part is then a ”component of in the basis . Under an inertial symmetry transformation, they transform in the usual way
where is the matrix representation of the symmetry group. We assume, that the representations is irreducible so that the particle has a unique charge or isotopic spin. The decomposition in Eq. (16) is particularly useful because it will allow us to interpret the effect of the external potential field on the particle as a precession of the internal basis.
Now, when the test particle moves from to through the external potential field, changes by an amount given
In general, must contain both the change in the external -dependent part of and change in the internal space basis . From Eq. (16), we can expand to first order in as
The second terms contains the change in the internal space basis. This term is given by the connection which we discussed above; it describes the effect of the external potential field on the internal space direction of the particle.
We now need to calculated the change in the internal space basis. The connection between the internal space direction at different space-time points is given by an internal rotation. In this case, the internal directions are specified by a set of basis vectors, so we must calculate the change from an infinitesimal internal rotation which is associated with the external displacement .
From Eq. (14), we calculate the infinitesimal internal rotation ,
which rotates the internal basis by an amount ,
The generators act like matrix operators on the column basis vector so we can write
Expanding to first order in , we obtain
which then gives for the change in the basis,
The net change will give us the connection that we have been seeking. Let us therefore introduce the new ”connection operator”
We thus finally obtain for the total change ,
where we have put in in order to factor out the basis vector . Now, we can factor the change into its own external and internal parts
The operator the gauge covariant derivative which describes the changes in both the external and internal parts of . Thus we get from Eq. (27)
For the case of the electromagnetic gauge group , the internal space is one dimensional so that Eq. (29) reduces to
This is the ”canonical momentum” which is familiar from Electromagnetism. We can also deduce from the example of the group, that the connection operator defined in Eq. (26), should be identified as the generalized version of the vector potential field . Thus, we conclude that the external potential field is indeed a connection in the internal symmetry space.
ii.9 Choosing a gauge
We know that the usual reason for selecting a particular gauge is to simplify a calculation or to explicitly display an interesting features of a problem. This freedom to choose a gauge is another example of the arbitrariness in the vector potential.
The choice of a gauge actually involves both gauge invariance and Lorentz invariance simultaneously. A particular gauge usually imposes a constraints on the vector potential, such as for the Coulomb gauge. In general, such equations are obviously not Lorentz invariant. Yet we know that two observers, each in a different inertial frame, can choose the Coulomb gauge for the same electromagnetism problem. The electric and magnetic fields observed in the two different frames can then be related by the usual Lorentz transformations between two frames. This example points out the fact that the space-time location at which the internal coordinate is evaluated is also not a fixed position. A Lorentz transformation which changes the spatial coordinate in the inertial frame also affects the value of the internal angles ad can be interpreted as an internal rotation. Thus, regardless of whether the coordinate change is associated with Lorentz transformation of observers or an actual movement of the particle in the external field, the effect on the internal space is the same: it is rotated by a gauge transformation.
A Lorentz transformation between two inertial frames is therefore always associated with a gauge transformation. Thus, the vector potential observed in the two frames are related by
where is the Lorentz transformation. This shows that the vector potential actually does not transform like an ordinary four-vector under a Lorentz transformation. It picks up an extra term due to the rotaion in the internal space. This interesting fact is well known in quantum field theory but it is rarely mentioned in ordinary electromagnetism bjorken ().
We can now see exactly what is involved for a particular choice of gauge. In the Lorentz gauge, is required to be invariant,
even though is not a true four-vector. From Eq. (31), we see that this is possible only if
which is the familiar equation for in the Lorentz gauge. Thus we see that the choice of Lorentz gauge is a requirement that the effect of the internal space precession be eliminated so that can be treated as if it were a relativistic invariant. By the same reasoning, other gauges like the Coulomb gauge are not invariant because the gauge condition does not completely eliminate the extra internal precession term. Thus in the Coulomb gauge
but and are not related by a simple Lorentz transformation. An additional gauge rotation is required france ().
Iii Gravity from gauge invariants
The above considerations on gauge symmetry can be developed for gravity. In this section, before discussing the physical meaning of GR, we will derive in a very general way the field equations considering manifolds equipped with curvature and torsion. Specifically, we want to show the role of global and local Poincaré invariance and the relevance of conservation laws in any theory of gravity. The approach is completely general and suitable for spinor, vector, bivector and tetrad fields independently of their specific physical meaning. Our first issue is what can ”generate” gravity.
iii.1 What can ”generate” gravity?
Since the perturbative scheme is unsatisfactory because it fails over one loop level and cannot be renormalized book (), as we have seen above, we can ask what can to produce gravity or in other words if there exists invariance principles leading to the gravitation SF (). Following the prescriptions of GR, the physical space-time is assumed to be a four-dimensional differential manifold. In Special Relativity, this manifold is the Minkwoski flat-space-time while, in GR, the underlying space-time is assumed to be curved in order to describe the effects of gravitation.
Utiyama Utiyama () was the first to propose that GR can be seen as a gauge theory based on the local Lorentz group in much the same way that the Yang-Mills gauge theory Yang-Mills () was developed on the basis of the internal isospin gauge group . In this formulation the Riemannian connection is the gravitational counterpart of the Yang-Mills gauge fields. While , in the Yang-Mills theory, is an internal symmetry group, the Lorentz symmetry represents the local nature of space-time rather than internal degrees of freedom. The Einstein Equivalence Principle, asserted for GR, requires that the local space-time structure can be identified with the Minkowski space-time possessing Lorentz symmetry. In order to relate local Lorentz symmetry to the external space-time, we need to solder the local space to the external space. The soldering tools are the tetrad fields. Utiyama regarded the tetrads as objects given a priori. Soon after, Sciama Sciama () recognized that space-time should necessarily be endowed with torsion in order to accommodate spinor fields. In other words, the gravitational interaction of spinning particles requires the modification of the Riemann space-time of GR to be a (non-Riemannian) curved space-time with torsion. Although Sciama used the tetrad formalism for his gauge-like handling of gravitation, his theory fell shortcomings in treating tetrad fields as gauge fields. Kibble Kibble () made a comprehensive extension of the Utiyama gauge theory of gravitation by showing that the local Poincaré symmetry ( represents the semi-direct product) can generate a space-time with torsion as well as curvature. The gauge fields introduced by Kibble include the tetrads as well as the local affine connection. There have been a variety of gauge theories of gravitation based on different local symmetries which gave rise to several interesting applications in theoretical physics Grignani (); hehlrev2 (); Inomata (); Ivanov1 (); Ivanov2 (); Mansouri1 (); Mansouri2 (); Ivanenko (); Sardanashvily (); Chang ().
Following the Kibble approach, it can be demonstrated how gravitation can be formulated starting from a pure gauge point of view. In particular, the aim of this section is to show, in details, how a theory of gravitation is a gauge theory which can be obtained starting from the local Poincaré symmetry and this feature works not only for GR but also for ETGs.
A gauge theory of gravity based on a nonlinear realization of the local conformal-affine group of symmetry transformations can be formulated in any case ali (). This means that the coframe fields and gauge connections of the theory can be always obtained. The tetrads and Lorentz group metric have been used to induce a space-time metric. The inhomogenously transforming (under the Lorentz group) connection coefficients give rise to gravitational gauge potentials used to define covariant derivatives accommodating minimal coupling of matter and gauge fields. On the other hand, the tensor valued connection forms can be used as auxiliary dynamical fields associated with the dilation, special conformal and deformation (shear) degrees of freedom inherent to the bundle manifold. This allowes to define the bundle curvature of the theory. Then boundary topological invariants have been constructed. They served as a prototype (source free) gravitational Lagrangian. Finally the Bianchi identities, covariant field equations and gauge currents are obtained.
Here, starting from the Invariance Principle, we consider first the Global Poincaré Invariance and then the Local Poincaré Invariance. This approach lead to construct a given theory of gravity as a gauge theory. This point of view, if considered in detail, can avoid many shortcomings and could be useful to formulate self-consistent schemes for any theory of gravity.
iii.2 Invariance Principles and the Noether Theorem
As it is well-known, field equations and conservation laws can be obtained from a least action principle. The same principle is the basis of any gauge theory so we start from it to develop our considerations. Let us start from a least action principle and the Noether theorem.
Let be a multiplet field defined at a space-time point and , ; be the Lagrangian density of the system. In order to make a distinction between the global transformations and the local transformations, for the moment we use the Latin indices () for the former and the greek indices () for the latter. The action integral of the system over a given space-time volume is defined by
Now let us consider the infinitesimal variations of the coordinates
and the field variables
Correspondingly, the variation of the action is given by
Since the Jacobian for the infinitesimal variation of coordinates becomes
the variation of the action takes the form,
For any function of , it is convenient to define the fixed point variation by,
Expanding the function to first order in as
The advantage to have the fixed point variation is that commutes with :
For , we have
Using the fixed point variation in the integrand of (41) gives
If we require the action integral defined over any arbitrary region be invariant, that is, , then we must have
If , then , that is, the Lagrangian density is invariant. In general, however, , and transforms like a scalar density. In other words, is a Lagrangian density unless .
For convenience, let us introduce a function that behaves like a scalar density, namely
We further assume . Then we see that
Hence the action integral remains invariant if
The newly introduced function is the scalar Lagrangian of the system.
Let us calculate the integrand of (50) explicitly. The fixed point variation of is a consequence of a fixed point variation of the field ,
which can be cast into the form,
Consequently, we have the action integral in the form
is the canonical energy-momentum tensor density. If the variations are chosen in such a way that over and vanishes on the boundary of , then gives us the Euler-Lagrange equation,
On the other hand, if the field variables obey the Euler-Lagrange equation, , then we have
which gives rise, considering also the Noether theorem, to conservation laws. These very straightforward considerations are at the basis of our following discussion.
iii.3 The Global Poincaré invariance
As standard, we assert that our space-time in the absence of gravitation is a Minkowski space . The isometry group of is the group of Poincaré transformations (PT) which consists of the Lorentz group , and the translation group , . The Poincaré transformations of coordinates are
where and are real constants, and satisfy the orthogonality conditions . For infinitesimal variations,
where . While the Lorentz transformation forms a six parameter group, the Poincaré group has ten parameters. The Lie algebra for the ten generators of the Poincaré group is
where are the generators of Lorentz transformations, and are the generators of four-dimensional translations. Obviously, for the Poincaré transformations (62). Therefore, our Lagrangian density , which is the same as with in this case, is invariant; namely, for .
Suppose that the field transforms under the infinitesimal Poincar é transformations as
where the tensors are the generators of the Lorentz group, satisfying
Correspondingly, the derivative of transforms as
Since the choice of infinitesimal parameters and is arbitrary, the vanishing variation of the Lagrangian density leads to the identities,
We also obtain the following conservation laws
These conservation laws imply that the energy-momentum and angular momentum, respectively
are conserved. This means that the system invariant under the ten parameter symmetry group has ten conserved quantities. This is an example of Noether symmetry. The first term of the angular momentum integral corresponds to the spin angular momentum while the second term gives the orbital angular momentum. The global Poincaré invariance of a system means that, for the system, the space-time is homogeneous (all space-time points are equivalent) as dictated by the translational invariance and is isotropic (all directions about a space-time point are equivalent) as indicated by the Lorentz invariance. It is interesting to observe that the fixed point variation of the field variables takes the form
We remark that are the generators of the Lorentz transformation and are those of the translations.
iii.4 The Local Poincaré invariance
As next step, let us consider a modification of the infinitesimal Poincaré transformations (63) by assuming that the parameters and are functions of the coordinates and by writing them altogether as
which we call the local Poincaré transformations (or the general coordinate transformations). As before, in order to distinguish between global transformations and local transformation, we use the Latin indices , , , , for the former and the Greek indices , , , , for the latter. The variation of the field variables defined at a point is still the same as that of the global Poincaré transformations,
The corresponding fixed point variation of takes the form,
Differentiating both sides of (76) with respect to , we have
By using these variations, we obtain the variation of the Lagrangian ,
which is no longer zero unless the parameters and become constants. Accordingly, the action integral for the given Lagrangian density is not invariant under the local Poincaré transformations. We notice that while for the local Poincaré transformations, does not vanish under local Poincaré transformations. Hence, as expected is not a Lagrangian scalar but a Lagrangian density. As mentioned earlier, in order to define the Lagrangian , we have to select an appropriate non-trivial scalar function satisfying
Now we consider a minimal modification of the Lagrangian so as to make the action integral invariant under the local Poincaré transformations. It is rather obvious that if there is a covariant derivative which transforms as
then a modified Lagrangian , , , , , obtained by replacing of , , by , remains invariant under the local Poincaré transformations, that is
To find such a -covariant derivative, we introduce the gauge fields and define the -covariant derivative
in such a way that the covariant derivative transforms as
The transformation properties of are determined by and . Making use of
and comparing with (82) we obtain,